Analogue of a Fock-type integral arising from electromagnetism and its applications in number theory

Abstract

Closed-form evaluations of certain integrals of \(J_{0}(\xi )\), the Bessel function of the first kind, have been crucial in the studies on the electromagnetic field of alternating current in a circuit with two groundings, as can be seen from the works of Fock and Bursian, Schermann, etc. Koshliakov’s generalization of one such integral, which contains \(J_s(\xi )\) in the integrand, encompasses several important integrals in the literature including Sonine’s integral. Here, we derive an analogous integral identity where \(J_{s}(\xi )\) is replaced by a kernel consisting of a combination of \(J_{s}(\xi )\), \(K_{s}(\xi )\) and \(Y_{s}(\xi )\). This kernel is important in number theory because of its role in the Voronoï summation formula for the sum-of-divisors function \(\sigma _s(n)\). Using this identity and the Voronoï summation formula, we derive a general transformation relating infinite series of products of Bessel functions \(I_{\lambda }(\xi )\) and \(K_{\lambda }(\xi )\) with those involving the Gaussian hypergeometric function. As applications of this transformation, several important results are derived, including what we believe to be a corrected version of the first identity found on page 336 of Ramanujan’s Lost Notebook.

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Notes

  1. 1.

    These conditions, which are more general than the ones given in [44, p. 416, Equation (2)] and [21, p. 693, Formula 6.596.7], are established in Sect. 4.

  2. 2.

    Note that \(\epsilon ^{-\left( \nu -1\right) }=\overline{\epsilon }^{\left( \nu -1\right) }\), so there is no possibility of confusion unless, of course, one does not put the parentheses. However, we wish to keep the results in the sequel symmetric with respect to \(\epsilon \) and \(\overline{\epsilon }\) whenever we can. We will use, without mention, the fact that \(\overline{\epsilon }=1/\epsilon \).

  3. 3.

    We have inserted some comments in square brackets and changed some notations to conform them with those used in our paper.

  4. 4.

    There is no condition on a specified in [7]; however, the result can be seen to be true for \(-\pi<\arg (a)<\pi \).

  5. 5.

    There are two typos in Koshliakov’s version of (2.9), namely, in his version, the arguments of the K-Bessel functions inside the integral on the left contain a, and those on the right contain x. Both these typos are corrected, albeit with renaming of his parameters, in our version of his formula given in (2.9).

  6. 6.

    As mentioned in [12], the first equality in (6.17) is an obvious modification of [44, p. 410, Equation (13.45)].

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Acknowledgements

The authors are really grateful to the anonymous referees for several important suggestions which improved the quality of the paper. They also sincerely thank Professor Anna Vishnyakova from V. N. Karazin Kharkiv National University for translating for them the last three pages of [17]. They also sincerely thank Karrie Peterson, Head, MIT Libraries, for sending them a copy of [38], and Nico M. Temme for informing them of the reference [13]. The first author’s research is supported by the SERB-DST grant ECR/2015/000070. He sincerely thanks SERB-DST for the support. The second author’s research is partially supported by funds provided by the University of North Carolina at Charlotte (UNCC). Part of this work was done while the first author was visiting UNCC in the summers of 2018 and 2019. He thanks UNCC for the wonderful hospitality.

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Correspondence to Atul Dixit.

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Dixit, A., Roy, A. Analogue of a Fock-type integral arising from electromagnetism and its applications in number theory. Res Math Sci 7, 25 (2020). https://doi.org/10.1007/s40687-020-00223-6

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Keywords

  • Bessel functions
  • Generalized sum-of-divisors function
  • Voronoï summation formula
  • Analytic continuation

Mathematics Subject Classification

  • Primary 11M06
  • 33E20
  • Secondary 33C10